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Abstract

As shown in a recent letter [Nature 452, 728 (2008)
] with a microscopic model, the phenomenon of the extraordinary optical transmission (EOT) is intrinsically due to two distinct surface waves: the surface plasmon polariton and the quasi-cylindrical wave (quasi-CW) that efficiently funnel light into the hole aperture at resonance. Here we present a comprehensive microscopic model of the EOT that takes into account the two surface waves. The model preserves the desirable physical insight of the previous approach, but since it additionally takes into account the quasi-CWs, it provides highly accurate predictions over a much broader spectral range, from visible to microwave radiation. The net outcome is a complete understanding of many aspects of the EOT and especially of the role of the metal conductivity that has largely puzzled the initial interpretations. We believe that the main conclusions of the present analysis may be applied to many Wood-type surface resonances on metallic surfaces.

Figures (5)

Fig. 1 Elementary HW scattering events (a)–(c) for building up the EOT phenomenon (d). (a)–(c) HW scattering coefficients at a 1D hole chain under illumination (a) by a HW, (b) by the fundamental mode of the chain, and (c) by a TM-polarized plane wave. This defines six scattering coefficients, ρ, τ, α, β(kx), r, and t(kx), with kx being the x-component of the wave vector of the incident or scattered plane waves. (d) Modal scattering coefficients used in the classical Fabry–Perot equation of the EOT. The arrows on the surface in the chain and in free space denote HWs, fundamental chain modes, and plane waves, respectively. The arrows denoting incident and scattered waves are in red and in green, respectively.

Fig. 2 Coupled-wave coefficients of a nonperiodic array of 1D hole chains illuminated by a TM-polarized plane wave at oblique incidence. The notation of arrows follows that in Fig. 1. Pn, Qn, and cn denote the coefficients of the right-going HW, the left-going HW, and the down-going fundamental chain mode that originate from the nth chain at x=xn(n=1,2,…,N).

Fig. 3 Comparison between the RCWA data and the model predictions for different incident angles θ and for the near-infrared band. All the data are obtained for a gold membrane in air perforated by a periodic 2D array of square holes; the period is a=0.94μm, the hole side length is D=0.266μm, and the membrane thickness is d=0.2μm. (a) Zeroth-order transmittance obtained with the RCWA (left), the SPP model (middle), and the HW model (right). The dotted-white lines represent the air light lines, at which a diffraction order propagates parallel to the metal surface. (b),(c) Zeroth-order transmittance and reflectance spectra for two incident angles θ=0° (red) and 5° (green), obtained with the RCWA (dotted), the SPP model (dashed), and the HW model (solid). The inset in (b) shows the transmittance in a logarithmic scale and evidences the existence of a deep transmission minimum.

Fig. 4 Comparison between the fully vectorial data and the model predictions for various wavelength ranges. All the data are obtained for a gold membrane in air perforated by a periodic 2D array of square holes; the hole side length is D/a=0.28, and the membrane thickness is d/a=0.21, with a being the grating period. (a),(b) Zeroth-order transmittance and reflectance spectra under normal incidence, which are obtained with the RCWA (dotted), the SPP model (dashed), and the HW model (solid) and are shown in the visible (red, a=0.68μm), the near-infrared (green, a=0.94μm), and the thermal-infrared (blue, a=2.92μm) bands. (c),(d) Perfect conductor results under normal and oblique incidence (θ=5°), which show the zeroth-order transmittance T (red) and reflectance R (green) spectra for very low frequencies. The fully vectorial data are shown with dotted curves and the HW-model predictions with dot marks. The inset in (c) shows T in a logarithmic scale.

Fig. 5 Phase-matching condition under normal incidence. (a) The phase (upper) and the modulus (lower) of τ+ρ (dotted-red lines), 1/ΣHSP+1=exp(−ikSPa) (dashed-green lines), and 1/ΣHHW+1 (solid-blue lines). (b) Transmittance |tA|2 obtained with the RCWA (dotted-red lines), the pure SPP model (dashed-green lines), and the HW model (solid-blue lines). The inset shows |tA|2 in a logarithmic scale. The phase-matching condition, which corresponds to the two intersections in (a) for predicting the peak wavelength of |tA|2, is labeled by the left and right dashed-dotted vertical lines for the SPP model and for the HW model, respectively.